Equal Tangents from a Point

Statement

If two tangents are drawn to a circle from the same external point, then the lengths of the tangents are equal. Symbolically:

\[ PA = PB \]

where \(PA\) and \(PB\) are tangents from point \(P\) to the circle at points \(A\) and \(B\).

Why it’s true

• Draw radii \(OA\) and \(OB\) to the tangent points.
• \(\angle OAP = \angle OBP = 90^\circ\) because a radius is perpendicular to a tangent at the point of contact.
• Triangles \(\triangle OAP\) and \(\triangle OBP\) are congruent (right angle, common hypotenuse \(OP\), and radius \(OA=OB\)).
• By congruence, \(PA = PB\).

Recipe (how to use it)

1. Identify the external point and the two tangents drawn to the circle.
2. Apply the rule: the tangents from the same external point are equal in length.
3. Use this equality to solve for unknown lengths in geometry problems.

Spotting it

Look for problems where a circle and tangents from an external point are mentioned. The key phrase is “tangents from the same point”.

Common pairings

• Circle theorems involving tangents and radii.
• Right-angled triangles formed by radii and tangents.
• Geometry proofs requiring equal lengths.

Mini examples

1. Given: A point \(P\) outside a circle has tangents \(PA\) and \(PB\). If \(PA=8\), then \(PB=8\).
2. Given: From point \(P\), tangents \(PA\) and \(PB\) touch a circle of radius 5 cm. If \(OP=13\), then \(PA=PB=\sqrt{13^2-5^2} = 12\).

Pitfalls

• Confusing tangents with secants (tangents touch once, secants cut twice).
• Forgetting that both tangents from the same point are equal, not just similar.

Exam strategy

• Always mark the two tangents as equal when drawn from the same point.
• Use congruent triangles (OAP and OBP) to justify in proofs.
• Check whether Pythagoras can be applied in the right-angled triangles formed.

Summary

Tangents drawn to a circle from a single external point are equal in length. This is a standard circle theorem and is useful for proving equal lengths and solving geometry problems.